Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $2\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8N0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}5&16\\4&87\end{bmatrix}$, $\begin{bmatrix}5&94\\62&57\end{bmatrix}$, $\begin{bmatrix}37&40\\20&43\end{bmatrix}$, $\begin{bmatrix}67&102\\10&11\end{bmatrix}$, $\begin{bmatrix}115&2\\74&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.g.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $368640$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^2}\cdot\frac{(5x+4y)^{48}(2575x^{8}+35200x^{7}y+222400x^{6}y^{2}+332800x^{5}y^{3}-376320x^{4}y^{4}-655360x^{3}y^{5}+901120x^{2}y^{6}+1310720xy^{7}+458752y^{8})^{3}(3675x^{8}+4800x^{7}y-46400x^{6}y^{2}+153600x^{5}y^{3}+1415680x^{4}y^{4}+2211840x^{3}y^{5}+901120x^{2}y^{6}+196608y^{8})^{3}}{(5x+4y)^{48}(5x^{2}+8xy+16y^{2})^{8}(5x^{4}-160x^{3}y-480x^{2}y^{2}+256y^{4})^{4}(55x^{4}+240x^{3}y-480x^{2}y^{2}-1280xy^{3}-256y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 60.48.0-20.b.1.2 | $60$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-20.b.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.192.3-40.d.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-40.e.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-40.h.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-40.i.1.8 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-40.bg.1.2 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-40.bh.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-40.bk.1.8 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-40.bl.1.4 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-120.cy.1.15 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-120.cz.1.14 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-120.dg.1.15 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-120.dh.1.14 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-120.gq.1.14 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-120.gr.1.15 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-120.gy.1.16 | $120$ | $2$ | $2$ | $3$ |
| 120.192.3-120.gz.1.16 | $120$ | $2$ | $2$ | $3$ |
| 120.288.8-120.bo.1.58 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-120.bz.1.54 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-40.m.1.8 | $120$ | $5$ | $5$ | $16$ |